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TitleUsing Quadratic Simplicial Elements for Hierarchical Approximation and Visualization (In Proceedings)
inVisualization and Data Analysis 2002, Proceedings
Author(s) David F. Wiley, Hank Childs, Bernd Hamann, Ken Joy, Nelson Max
Editor(s) Robert F. Erbacher, Philip C. Chen, Matti Groehn, Jonathan C. Roberts, Craig M. Wittenbrink
Keyword(s)quadratic elements, higher-order finite elements, approximation, hierarchical approximation, data-dependent, approximation, visualization, spline, refinement, multiresolution
Year 2002
Volume4665
PublisherSPIE
AddressBellingham, Washington
Pages32--43
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Abstract Best quadratic simplicial spline approximations can be computed, using quadratic Bernstein-Bézier basis functions, by identifying and bisecting simplicial elements with largest errors. Our method begins with an initial triangulation of the domain; a best quadratic spline approximation is computed; errors are computed for all simplices; and simplices of maximal error are subdivided. This process is repeated until a user-specified global error tolerance is met. The initial approximations for the unit square and cube are given by two quadratic triangles and five quadratic tetrahedra, respectively. Our more complex triangulation and approximation method that respects field discontinuities and geometrical features allows us to better approximate data. Data is visualized by using the hierarchy of increasingly better quadratic approximations generated by this process. Many visualization problems arise for quadratic elements. First tessellating quadratic elements with smaller linear ones and then rendering the smaller linear elements is one way to visualize quadratic elements. Our results show a significant reduction in the number of simplices required to approximate data sets when using quadratic elements as compared to using linear elements.